Question

How do you write an nth term rule for `1,-4,16,-64` and find `a_6`?

Answer 1

`a_6 = 4096` or `a_6 = -1024` depending on convention.

Explanation:

The series here looks to be:
`sum_(n=0) (-4)^n`

This does present a slight ambiguity in the way the question was asked. That is, if we assume that we start counting from `n=0` then:
`a_n = (-4)^n`

However if we want to start counting from `n=1` then we can write the formula as:
`a_n = (-4)^(n-1)`

The later formula is advantageous in that the `n`th term actually corresponds to that `n`. That is, in the first equation, the third term is at `a_2` as opposed to `a_3`. In the second equation, the `n`th term is denoted simply by `a_n`.

Thus, it is not completely clear what the correct answer would be for `a_6`.

It may be:
`a_6 = 4096`
if we started counting from `n=0`

Or
`a_6 = -1024`
if we started counting from `n = 1`

We'll note that the first answer is actually the 6th term in the series, while the second is actually the 5th. If this problem were presented on homework I would most likely ask for clarification from the instructor.